Works by Wolfgang Rautenberg

Traditional logic as a part of philosophy is one of the oldest scientific disciplines. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, Russell and others to create a logistic foundation for mathematics. It steadily developed during the 20th century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy. While there are already several well-known textbooks on mathematical logic, this book is unique in that it is (...) much more concise than most others, and the material is treated in a streamlined fashion which allows the professor to cover many important topics in a one semester course. Although the book is intended for use as a graduate text, the first three chapters could be understood by undergraduates interested in mathematical logic. These initial chapters cover just the material for an introductory course on mathematical logic combined with the necessary material from set theory. This material is of a descriptive nature, providing a view towards decision problems, automated theorem proving, non-standard models and other subjects. The remaining chapters contain material on logic programming for computer scientists, model theory, recursion theory, Godel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. The author has provided exercises for each chapter, as well as hints to selected exercises. About the German edition: …The book can be useful to the student and lecturer who prepares a mathematical logic course at the university. What a pity that the book is not written in a universal scientific language which mankind has not yet created. - A.Nabebin, Zentralblatt. (shrink)

It is shown that the class of reduced matrices of a logic is a 1 st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 st order.

We provide a finite axiomatization of the consequence , i.e. of the set of common sequential rules for and . Moreover, we show that has no proper non-trivial strengthenings other than and . A similar result is true for , but not, e.g., for +.

We provide a finite axiomatization of the consequence $\vdash ^{\wedge}\cap \vdash ^{\vee}$ , i.e. of the set of common sequential rules for $\wedge $ and $\vee $ . Moreover, we show that $\vdash ^{\wedge}\cap \vdash ^{\vee}$ has no proper non-trivial strengthenings other than $\vdash ^{\wedge}$ and $\vdash ^{\vee}$ . A similar result is true for $\vdash ^{\leftrightarrow}\cap \vdash ^{\rightarrow}$ , but not, e.g., for $\vdash ^{\leftrightarrow}\cap \vdash ^{+}$.

We show (1) the consequence determined by a variety V of algebraic semigroup matrices is finitely based iffV is finitely based, (2) the consequence determined by all 2-valued semigroup connectives, Λ, ∨, ↔, +, in other words the collection of common rules for all these connectives, is finitely based. For possible applications see Sect. 0.

Section 1 contains a Kripke-style completeness theorem for arbitrary intermediate consequences. In Section 2 we apply weak Kripke semantics to splittings in order to obtain generalized axiomatization criteria of the Jankov-type. Section 3 presents new and short proofs of recent results on implicationless intermediate consequences. In Section 4 we prove that these consequences admit no deduction theorem. In Section 5 all maximal logics in the 3 rd counterslice are determined. On these results we reported at the 1980 meeting on Mathematical (...) Logic at Oberwolfach. This paper concerns propositional logic only. (shrink)

Sections 1, 2 and 3 contain the main result, the strong finite axiomatizability of all 2-valued matrices. Since non-strongly finitely axiomatizable 3-element matrices are easily constructed the result reveals once again the gap between 2-valued and multiple-valued logic. Sec. 2 deals with the basic cases which include the important F i from Post's classification. The procedure in Sec. 3 reduces the general problem to these cases. Sec. 4 is a study of basic algebraic properties of 2-element algebras. In particular, we (...) show that equational completeness is equivalent to the Stone-property and that each 2-element algebra generates a minimal quasivariety. The results of Sec. 4 will be applied in Sec. 5 to maximality questions and to a matrix free characterization of 2-valued consequences in the lattice of structural consequences in any language. Sec. 6 takes a look at related axiomatization. problems for finite algebras and matrices. We study the notion of a propositional consequence with equality and, among other things, present explicit axiomatizations of 2-valued consequences with equality. (shrink)

Most material below is ranked around the splittings of lattices of normal modal logics. These splittings are generated by nite subdirect irreducible modal algebras. The actual computation of the splittings is often a rather delicate task. Rened model structures are very useful to this purpose, as well as they are in many other respects. E.g. the analysis of various lattices of extensions, like ES5, ES4:3 etc becomes rather simple, if rened structures are used. But this point will not be touched (...) here. The variety T BA , which corresponds to S4, is congruence- distributive. Hence any every tabular extension L S4 has nite many extensions only. Around 1975 it has been proved by several authors, that also the converse is true: If L S4 has nitely many extensions only, then L is necessarely tabular. These and other results will be extended to much richer lattices. For simplicity we state the results explicitely only for the lattice N of modal logics with one modal operator. But most of them carry over literally to the lattice Nk of k-ramied normal modal logics . It is easy to construct examples of 2-ramied modal logics which are P OST-complete but not tabular. Whether this will be possible for normal modal logic seems to be an open problem. In view of results below it seems very unlikely that such examples exist. (shrink)

We are concerned with modal logics in the class EM0 of extensions of M0 . G denotes re exive frames. MG the modal logic on G in the sense of Kripke. M is nite if M = MG for some nite G. Finite G's will be drawn as framed diagrams, e.g. G = ! ; G = ! ; the latter shorter denoted by . EM0 is a complete lattice with zero M0 and one M . If M M0 M0 (...) is a succ of M. An ip of M is an immediate predecessor of M. E.g. M ! and M are the only ip's of M in ES4. M[P] denotes the extension of M adding P as a \new axiom", e.g. S4 = M0 [p ! p]. M is nitely axiomatizable if M = M0 [P] for some formula P. One easily shows if M is f.a. then each predecessor is separated from M by an ip of M. It is known that each nite M is f.a. and has only nitely many succ's all of which are nite. (shrink)